Amin -

I am guessing that you have 'outlier' problems?  I'm guessing you might mean least absolute deviation regression when you say "robust?"  Both GLS and least absolute deviation regression can help with that.  The idea is not to let an "odd" data point influence your regression too much. 

I'd say that the WLS (weighted least squares) part of GLS may help best if you have a true pattern of heteroscedasticity naturally found in the error structure.  The absolute deviation regression might be better for a few stray unusual data points.

The following may be of some use to you:

 https://onlinecourses.science.psu.edu/stat501/node/353

You might want to compare residual analyses for the two methods. 

My exposure to panel data has been limited, but I think the above might address your issue. 

Cheers - Jim

 I would say, the one will give you better results to support your hypothesis. After all, that is what we usually want from our analysis results. 

If you've gone through all the pre-test of model specification and you've ended with fe and your models are suffering with both heteroskedasticity and auto-correlation, you can choose Robust.

However, if you get better results from GLS to support you hypothesis then you can go for GLS, but you need be clear at econometric base. 

Thank you Iftekhar Ahmad and James R Knaub for sharing your knowledge. 

Just to make sure: Do you ask about which econometric model to use, or how to control your standard errors? Because in panel data, it is most likely to have autocorrelation (and heteroscedasticity), and you can control for this by using robust and clustered standard errors. Depending on the software you are using, this is done pretty easy.  E.g. in stata, just use reg y x, cluster(id) , whereby cluster controls for both, autocorrelation on the time dimension of each individual and heteroscedasticity.

Jakob Schweter... thank you for your answer but actually i am curious that which method either robust/cluster or GLS is more appropriate in case FE panel data. because both tests are used to correct hetro and auto-correlation.  if we can use robust or cluster then when can we use GLS

The basic difference of GLS is in the variance of the estimates. The assumptions made about the error term of the model differ in GLS model. For instant, in OLS we assume that there are no off diagonal elements different from zero.But, in GLS we assume that Σ is the variance-covariance matrix. Thus, the rationale of choosing GLS over OLS is indeed to gain asymptotic efficiency, perhaps getting smaller variance for n.

I would suggest some readings for more clarification. 

https://homepage.ntu.edu.tw/~ckuan/pdf/et01/et_Ch4.pdf

http://http-server.carleton.ca/~bchu/Document/ECON5005/lecture05.pdf

https://www.tandfonline.com/doi/abs/10.1207/S15328007SEM0704_3?journalCode=hsem20

But with clustered standard errors you allow for the dependence as well, just limited for specific groups. And in panel data, it is very common to use basic OLS, demeanind the data (which is then FE), and cluster the data at the individual level to account for the autocorrelation in the data. The clustering (at least in stata and R), also controls for heteroscedasticity.

If you suspect a grouping into a second dimension, there is also the possibility for a twoway-cluster.

GLS, or more accurate, Feasible GLS is usually only applied if you have to include the lag dependent variable (dynamic term), because FE does not work in this case (which is complete different story then). 

Thank you Jakob Schweter

I agree with @Jakob, and will take Heteroskedasticity Robust Standard Errors.